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<p class='katex-block katex-error' title='ParseError: KaTeX parse error: No such environment: equation at position 7: \begin{̲e̲q̲u̲a̲t̲i̲o̲n̲}̲
\begin{split} …'>\begin{equation}
\begin{split}   I_n &amp;=a+b \cos(\varphi+\Delta\varphi_n)\\
      &amp;=a+b\cos\varphi\cos\Delta\varphi_n-b\sin\varphi\sin\Delta\varphi_n
\end{split}
\end{equation}
</p>
<p>将<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mi>cos</mi><mo>⁡</mo><mi>φ</mi></mrow><annotation encoding="application/x-tex">b\cos\varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">φ</span></span></span></span>记作<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">c</span></span></span></span>，将<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mi>sin</mi><mo>⁡</mo><mi>φ</mi></mrow><annotation encoding="application/x-tex">b\sin\varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">φ</span></span></span></span>记作<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>，那么</p>
<p class='katex-block katex-error' title='ParseError: KaTeX parse error: No such environment: equation at position 7: \begin{̲e̲q̲u̲a̲t̲i̲o̲n̲}̲
I_n=a+c\cos\De…'>\begin{equation}
I_n=a+c\cos\Delta\varphi_n-s\sin\Delta\varphi_n
\end{equation}
</p>
<p>我们需要从N个公式（2）中求取未知量<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo separator="true">,</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a,c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span></span></span></span>和<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>,很自然地采用最小二乘法求解，最优化以下偏差：</p>
<p class='katex-block katex-error' title='ParseError: KaTeX parse error: No such environment: equation at position 7: \begin{̲e̲q̲u̲a̲t̲i̲o̲n̲}̲
\varepsilon=\s…'>\begin{equation}
\varepsilon=\sum_{n=1}^N \big(I_n-(a+c\cos\Delta\varphi_n-s\sin\Delta\varphi_n)\big)^2
\end{equation}
</p>
<p>由偏导<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>ε</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>a</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{\partial\varepsilon}{\partial a}=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251079999999999em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801079999999999em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathdefault mtight">a</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathdefault mtight">ε</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>可得</p>
<p class='katex-block katex-error' title='ParseError: KaTeX parse error: No such environment: equation at position 7: \begin{̲e̲q̲u̲a̲t̲i̲o̲n̲}̲
\sum_{n=1}^{N}…'>\begin{equation}
\sum_{n=1}^{N}I_n=aN+c\sum_{n=1}^{N}\cos\Delta\varphi_n-s\sum_{n=1}^{N}\sin\Delta\varphi_n
\end{equation}
</p>
<p>由偏导<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>ε</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>c</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{\partial\varepsilon}{\partial c}=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251079999999999em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801079999999999em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathdefault mtight">c</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathdefault mtight">ε</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>可得</p>
<p class='katex-block katex-error' title='ParseError: KaTeX parse error: No such environment: equation at position 7: \begin{̲e̲q̲u̲a̲t̲i̲o̲n̲}̲
\sum_{n=1}^{N}…'>\begin{equation}
\sum_{n=1}^{N}I_n\cos\Delta\varphi_n=a\sum_{n=1}^{N}\cos\Delta\varphi_n+c\sum_{n=1}^{N}\cos^2\Delta\varphi_n-s\sum_{n=1}^{N}\sin\Delta\varphi_n\cos\Delta\varphi_n
\end{equation}
</p>
<p>由偏导<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>ε</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>s</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{\partial\varepsilon}{\partial s}=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2251079999999999em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801079999999999em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathdefault mtight">ε</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>可得</p>
<p class='katex-block katex-error' title='ParseError: KaTeX parse error: No such environment: equation at position 7: \begin{̲e̲q̲u̲a̲t̲i̲o̲n̲}̲
\sum_{n=1}^{N}…'>\begin{equation}
\sum_{n=1}^{N}I_n\sin\Delta\varphi_n=a\sum_{n=1}^{N}\sin\Delta\varphi_n+c\sum_{n=1}^{N}\sin\Delta\varphi_n\cos\Delta\varphi_n-s\sum_{n=1}^{N}\sin^2\Delta\varphi_n
\end{equation}
</p>
<p>综合公式（4），公式（5）和公式（6），整理成矩阵形式，为</p>
<p class='katex-block katex-error' title='ParseError: KaTeX parse error: No such environment: equation at position 7: \begin{̲e̲q̲u̲a̲t̲i̲o̲n̲}̲
\begin{pmatrix…'>\begin{equation}
\begin{pmatrix}N&amp;\sum_{n=1}^{N}\cos\Delta\varphi_n&amp;-\sum_{n=1}^{N}\sin\Delta\varphi_n &amp; \\\sum_{n=1}^{N}\cos\Delta\varphi_n &amp; \sum_{n=1}^{N}\cos^2\Delta\varphi_n &amp; -\sum_{n=1}^{N}\sin\Delta\varphi_n\cos\Delta\varphi_n \\ \sum_{n=1}^{N}\sin\Delta\varphi_n &amp;  \sum_{n=1}^{N}\sin\Delta\varphi_n\cos\Delta\varphi_n &amp; -\sum_{n=1}^{N}\sin^2\Delta\varphi_n  \end{pmatrix}\begin{pmatrix}a\\c\\s\end{pmatrix}=\begin{pmatrix}\sum_{n=1}^{N}I_n\\\sum_{n=1}^{N}I_n\cos\Delta\varphi_n\\\sum_{n=1}^{N}I_n\sin\Delta\varphi_n\end{pmatrix}
\end{equation}
</p>
<p>由于<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><mi>cos</mi><mo>⁡</mo><mi mathvariant="normal">Δ</mi><msub><mi>φ</mi><mi>n</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\sum_{n=1}^{N}\cos\Delta\varphi_n=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2809409999999999em;vertical-align:-0.29971000000000003em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><mi>sin</mi><mo>⁡</mo><mi mathvariant="normal">Δ</mi><msub><mi>φ</mi><mi>n</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\sum_{n=1}^{N}\sin\Delta\varphi_n=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2809409999999999em;vertical-align:-0.29971000000000003em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>,<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><mi>sin</mi><mo>⁡</mo><mi mathvariant="normal">Δ</mi><msub><mi>φ</mi><mi>n</mi></msub><mi>cos</mi><mo>⁡</mo><mi mathvariant="normal">Δ</mi><msub><mi>φ</mi><mi>n</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\sum_{n=1}^{N}\sin\Delta\varphi_n\cos\Delta\varphi_n=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2809409999999999em;vertical-align:-0.29971000000000003em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><msup><mo><mi>cos</mi><mo>⁡</mo></mo><mn>2</mn></msup><mi mathvariant="normal">Δ</mi><msub><mi>φ</mi><mi>n</mi></msub><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\sum_{n=1}^{N}\cos^2\Delta\varphi_n=\frac{N}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2809409999999999em;vertical-align:-0.29971000000000003em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop">cos</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.217331em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.872331em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><msup><mo><mi>sin</mi><mo>⁡</mo></mo><mn>2</mn></msup><mi mathvariant="normal">Δ</mi><msub><mi>φ</mi><mi>n</mi></msub><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\sum_{n=1}^{N}\sin^2\Delta\varphi_n=\frac{N}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2809409999999999em;vertical-align:-0.29971000000000003em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.871868em;"><span style="top:-3.12076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.217331em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.872331em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, 代入公式（7）中可得，</p>
<p class='katex-block katex-error' title='ParseError: KaTeX parse error: No such environment: equation at position 7: \begin{̲e̲q̲u̲a̲t̲i̲o̲n̲}̲
\begin{pmatrix…'>\begin{equation}
\begin{pmatrix}N&amp;0&amp;0 \\0&amp;\frac{N}{2}&amp;0 \\ 0&amp;0&amp;-\frac{N}{2}\end{pmatrix}\begin{pmatrix}a \\ c \\ s \end{pmatrix}=\begin{pmatrix}\sum_{n=1}^{N}I_n\\\sum_{n=1}^{N}I_n\cos\Delta\varphi_n\\\sum_{n=1}^{N}I_n\sin\Delta\varphi_n\end{pmatrix}
\end{equation}
</p>
<p>此时，易得</p>
<p class='katex-block katex-error' title='ParseError: KaTeX parse error: No such environment: equation at position 7: \begin{̲e̲q̲u̲a̲t̲i̲o̲n̲}̲
\begin{split}
…'>\begin{equation}
\begin{split}
a&amp;=\frac1N\sum_{n=1}^{N}I_n\\
c&amp;=\frac2N\sum_{n=1}^{N}I_n\cos\Delta\varphi_n\\
s&amp;=-\frac2N\sum_{n=1}^{N}I_n\sin\Delta\varphi_n
\end{split}
\end{equation}
</p>
<p>由于<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mi>cos</mi><mo>⁡</mo><mi>φ</mi><mo>=</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">b\cos\varphi=c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">φ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">c</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo>=</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">b\sin\varphi=s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">φ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>,可得</p>
<p class='katex-block katex-error' title='ParseError: KaTeX parse error: No such environment: equation at position 7: \begin{̲e̲q̲u̲a̲t̲i̲o̲n̲}̲
\begin{split}
…'>\begin{equation}
\begin{split}
\tan\varphi&amp;=-\frac{\sum_{n=1}^{N}I_n\sin\Delta\varphi_n}{\sum_{n=1}^{N}I_n\cos\Delta\varphi_n}\\
b&amp;=\frac2N\sqrt{(\sum_{n=1}^{N}I_n\sin\Delta\varphi_n)^2+(\sum_{n=1}^{N}I_n\cos\Delta\varphi_n)^2}
\end{split}
\end{equation}
</p>
<p>推导完毕。</p>
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